Question

(x^3 dx)/(〖(5x〗^4+3)〖^5〗) решите интеграл

Answer 1

interpret ... (x^3 dx)/( ( [5x]^4+3) [^5] ) ...    $Rightarrow int{ frac{ x^3 dx }{ ( (5x)^4 + 3 )^5 } } = frac{1}{5^4} int{ frac{ (5x)^3 d (5x) }{ ( (5x)^4 + 3 )^5 } } =$

$= frac{1}{ 4 cdot 5^4 } int{ frac{1}{ ( (5x)^4 + 3 )^5 } } , cdot 4 cdot (5x)^3 d (5x) = frac{1}{ 4 cdot 5^4 } int{ frac{1}{ ( (5x)^4 + 3 )^5 } } , d (5x)^4 = \\ = frac{1}{ 4 cdot 5^4 } int{ frac{ d ( (5x)^4 + 3 ) }{ ( (5x)^4 + 3 )^5 } } = frac{1}{ 4 cdot 5^4 } int{ ( (5x)^4 + 3 )^{-5} } , d ( (5x)^4 + 3 ) = \\ = frac{1}{ 1 + [- 5] } cdot frac{1}{ 4 cdot 5^4 } cdot ( (5x)^4 + 3 )^{-4} + C = frac{1}{4} cdot frac{1}{ 4 cdot 5^4 } cdot frac{1}{ ( (5x)^4 + 3 )^4 } + C = \\ = frac{1}{ 2^2 cdot 2^2 cdot 5^4 } cdot frac{1}{ ( (5x)^4 + 3 )^4 } + C = frac{1}{ 2^4 cdot 5^4 } cdot frac{1}{ ( (5x)^4 + 3 )^4 } + C = frac{1}{ 10^4 } cdot frac{1}{ ( (5x)^4 + 3 )^4 } + C ;$

О т в е т :    $int{ frac{ x^3 dx }{ ( (5x)^4 + 3 )^5 } } = frac{1}{ 10 000 ( (5x)^4 + 3 )^4 } + C .$